Now suppose that $(n_1,n_2),(m_1,m_2)$ span $\mathbb{Z}^2$. Then $\exists a_1,b_1\in \mathbb{Z}$ such that $a_1(n_1,n_2)+b_1(m_1,m_2)=(1,0)$ and similarly $\exists a_2,b_2\in \mathbb{Z}$ such that $a_2(n_1,n_2)+b_2(m_1,m_2)=(0,1)$. So we have